Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [430,2,Mod(7,430)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(430, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([3, 10]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("430.7");
S:= CuspForms(chi, 2);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 430 = 2 \cdot 5 \cdot 43 \) |
Weight: | \( k \) | \(=\) | \( 2 \) |
Character orbit: | \([\chi]\) | \(=\) | 430.l (of order \(12\), degree \(4\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(3.43356728692\) |
Analytic rank: | \(0\) |
Dimension: | \(88\) |
Relative dimension: | \(22\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
7.1 | −0.707107 | + | 0.707107i | −0.695712 | + | 2.59643i | − | 1.00000i | −0.0995241 | − | 2.23385i | −1.34401 | − | 2.32790i | 0.420775 | + | 1.57036i | 0.707107 | + | 0.707107i | −3.65937 | − | 2.11274i | 1.64995 | + | 1.50920i | |
7.2 | −0.707107 | + | 0.707107i | −0.547703 | + | 2.04406i | − | 1.00000i | 2.11862 | − | 0.715150i | −1.05808 | − | 1.83265i | −0.838501 | − | 3.12933i | 0.707107 | + | 0.707107i | −1.28011 | − | 0.739071i | −0.992405 | + | 2.00378i | |
7.3 | −0.707107 | + | 0.707107i | −0.424244 | + | 1.58330i | − | 1.00000i | −1.40555 | − | 1.73909i | −0.819577 | − | 1.41955i | −0.632122 | − | 2.35911i | 0.707107 | + | 0.707107i | 0.271217 | + | 0.156587i | 2.22359 | + | 0.235851i | |
7.4 | −0.707107 | + | 0.707107i | −0.388695 | + | 1.45063i | − | 1.00000i | −0.951195 | + | 2.02367i | −0.750902 | − | 1.30060i | 0.957274 | + | 3.57260i | 0.707107 | + | 0.707107i | 0.644831 | + | 0.372294i | −0.758352 | − | 2.10355i | |
7.5 | −0.707107 | + | 0.707107i | −0.348214 | + | 1.29955i | − | 1.00000i | 1.37552 | + | 1.76294i | −0.672697 | − | 1.16515i | 0.277286 | + | 1.03484i | 0.707107 | + | 0.707107i | 1.03050 | + | 0.594958i | −2.21922 | − | 0.273949i | |
7.6 | −0.707107 | + | 0.707107i | 0.0121292 | − | 0.0452669i | − | 1.00000i | −2.07202 | + | 0.840686i | 0.0234319 | + | 0.0405852i | −0.477251 | − | 1.78112i | 0.707107 | + | 0.707107i | 2.59617 | + | 1.49890i | 0.870681 | − | 2.05959i | |
7.7 | −0.707107 | + | 0.707107i | 0.231077 | − | 0.862392i | − | 1.00000i | 2.23333 | − | 0.110550i | 0.446407 | + | 0.773199i | 0.738092 | + | 2.75460i | 0.707107 | + | 0.707107i | 1.90775 | + | 1.10144i | −1.50103 | + | 1.65738i | |
7.8 | −0.707107 | + | 0.707107i | 0.392168 | − | 1.46359i | − | 1.00000i | −2.07474 | − | 0.833933i | 0.757610 | + | 1.31222i | 1.23476 | + | 4.60819i | 0.707107 | + | 0.707107i | 0.609775 | + | 0.352054i | 2.05674 | − | 0.877385i | |
7.9 | −0.707107 | + | 0.707107i | 0.506805 | − | 1.89142i | − | 1.00000i | 0.195585 | − | 2.22750i | 0.979072 | + | 1.69580i | −0.522322 | − | 1.94933i | 0.707107 | + | 0.707107i | −0.722552 | − | 0.417165i | 1.43678 | + | 1.71338i | |
7.10 | −0.707107 | + | 0.707107i | 0.653568 | − | 2.43915i | − | 1.00000i | 1.81492 | + | 1.30616i | 1.26260 | + | 2.18688i | −0.0630011 | − | 0.235123i | 0.707107 | + | 0.707107i | −2.92423 | − | 1.68830i | −2.20694 | + | 0.359750i | |
7.11 | −0.707107 | + | 0.707107i | 0.867639 | − | 3.23807i | − | 1.00000i | −1.13496 | + | 1.92662i | 1.67615 | + | 2.90318i | 0.0566213 | + | 0.211314i | 0.707107 | + | 0.707107i | −7.13424 | − | 4.11896i | −0.559792 | − | 2.16486i | |
7.12 | 0.707107 | − | 0.707107i | −0.833140 | + | 3.10932i | − | 1.00000i | 1.35223 | + | 1.78086i | 1.60950 | + | 2.78774i | 1.22409 | + | 4.56838i | −0.707107 | − | 0.707107i | −6.37568 | − | 3.68100i | 2.21543 | + | 0.303091i | |
7.13 | 0.707107 | − | 0.707107i | −0.727882 | + | 2.71649i | − | 1.00000i | −2.12803 | − | 0.686647i | 1.40616 | + | 2.43554i | −0.979873 | − | 3.65693i | −0.707107 | − | 0.707107i | −4.25143 | − | 2.45457i | −1.99028 | + | 1.01921i | |
7.14 | 0.707107 | − | 0.707107i | −0.426111 | + | 1.59027i | − | 1.00000i | 1.81858 | + | 1.30107i | 0.823184 | + | 1.42580i | −0.688340 | − | 2.56892i | −0.707107 | − | 0.707107i | 0.250692 | + | 0.144737i | 2.20592 | − | 0.365937i | |
7.15 | 0.707107 | − | 0.707107i | −0.356683 | + | 1.33116i | − | 1.00000i | −2.17879 | − | 0.502846i | 0.689058 | + | 1.19348i | 0.982140 | + | 3.66540i | −0.707107 | − | 0.707107i | 0.953316 | + | 0.550397i | −1.89621 | + | 1.18507i | |
7.16 | 0.707107 | − | 0.707107i | −0.103919 | + | 0.387831i | − | 1.00000i | 1.79733 | − | 1.33027i | 0.200756 | + | 0.347719i | −0.783039 | − | 2.92234i | −0.707107 | − | 0.707107i | 2.45846 | + | 1.41939i | 0.330259 | − | 2.21154i | |
7.17 | 0.707107 | − | 0.707107i | −0.00223075 | + | 0.00832526i | − | 1.00000i | −1.58545 | + | 1.57682i | 0.00430947 | + | 0.00746423i | 0.306192 | + | 1.14273i | −0.707107 | − | 0.707107i | 2.59801 | + | 1.49996i | −0.00610084 | + | 2.23606i | |
7.18 | 0.707107 | − | 0.707107i | 0.0996077 | − | 0.371741i | − | 1.00000i | 2.14837 | − | 0.620072i | −0.192427 | − | 0.333294i | 1.10238 | + | 4.11412i | −0.707107 | − | 0.707107i | 2.46981 | + | 1.42594i | 1.08067 | − | 1.95759i | |
7.19 | 0.707107 | − | 0.707107i | 0.118826 | − | 0.443464i | − | 1.00000i | −0.834055 | − | 2.07469i | −0.229554 | − | 0.397599i | −0.174257 | − | 0.650336i | −0.707107 | − | 0.707107i | 2.41554 | + | 1.39461i | −2.05680 | − | 0.877264i | |
7.20 | 0.707107 | − | 0.707107i | 0.555341 | − | 2.07256i | − | 1.00000i | −1.79466 | + | 1.33387i | −1.07284 | − | 1.85821i | −1.21892 | − | 4.54906i | −0.707107 | − | 0.707107i | −1.38903 | − | 0.801955i | −0.325825 | + | 2.21220i | |
See all 88 embeddings |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
5.c | odd | 4 | 1 | inner |
43.d | odd | 6 | 1 | inner |
215.l | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 430.2.l.a | ✓ | 88 |
5.c | odd | 4 | 1 | inner | 430.2.l.a | ✓ | 88 |
43.d | odd | 6 | 1 | inner | 430.2.l.a | ✓ | 88 |
215.l | even | 12 | 1 | inner | 430.2.l.a | ✓ | 88 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
430.2.l.a | ✓ | 88 | 1.a | even | 1 | 1 | trivial |
430.2.l.a | ✓ | 88 | 5.c | odd | 4 | 1 | inner |
430.2.l.a | ✓ | 88 | 43.d | odd | 6 | 1 | inner |
430.2.l.a | ✓ | 88 | 215.l | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(430, [\chi])\).